EIT Mathematics and Statistics 2 — Questions and Answers
Question 1: The Laplace transform of f(t) = t·e^(2t) is:
- 1/(s-2)² (Correct answer)
- 1/(s+2)²
- s/(s-2)²
- 2/(s-2)²
Correct answer: 1/(s-2)²
Using the Laplace transform formula L{t·e^(at)} = 1/(s-a)², with a=2, the result is 1/(s-2)².
Question 2: Which numerical method uses the formula x_{n+1} = x_n - f(x_n)/f'(x_n) to find roots?
- Bisection method
- Newton-Raphson method (Correct answer)
- Secant method
- False position method
Correct answer: Newton-Raphson method
The Newton-Raphson method iteratively refines root estimates using the function and its derivative.
Question 3: A dataset has mean μ = 50 and standard deviation σ = 5. What percentage of data falls within one standard deviation of the mean, assuming a normal distribution?
- 50%
- 68% (Correct answer)
- 95%
- 99.7%
Correct answer: 68%
The empirical rule states that approximately 68% of data in a normal distribution falls within ±1σ of the mean.
Question 4: The determinant of the matrix [[3,1],[2,4]] equals:
- 10 (Correct answer)
- 14
- 8
- 12
Correct answer: 10
det = (3)(4) - (1)(2) = 12 - 2 = 10.
Question 5: Which of the following is the correct expression for the Taylor series expansion of e^x about x=0?
- Σ x^n/n! for n=0 to ∞ (Correct answer)
- Σ (-1)^n x^n/n! for n=0 to ∞
- Σ x^n/(2n)! for n=0 to ∞
- Σ x^(2n+1)/(2n+1)! for n=0 to ∞
Correct answer: Σ x^n/n! for n=0 to ∞
The Maclaurin series for e^x is 1 + x + x²/2! + x³/3! + … = Σ x^n/n!.
Question 6: Two events A and B are independent. If P(A) = 0.4 and P(B) = 0.5, what is P(A ∩ B)?
- 0.90
- 0.20 (Correct answer)
- 0.45
- 0.10
Correct answer: 0.20
For independent events, P(A ∩ B) = P(A) × P(B) = 0.4 × 0.5 = 0.20.
Question 7: What is the general solution to the differential equation dy/dx = 3y?
- y = Ce^(3x) (Correct answer)
- y = 3x + C
- y = Ce^(-3x)
- y = 3e^x + C
Correct answer: y = Ce^(3x)
Separating variables and integrating gives ln|y| = 3x + C₁, so y = Ce^(3x).
EIT Mathematics and Statistics 2
The Laplace transform of f(t) = t·e^(2t) is: